Abstract

We introduce a higher-order complex source that generates elegant Laguerre–Gaussian waves with radial mode number n and angular mode number m. We derive the integral and differential representations for the elegant Laguerre–Gaussian wave that in the appropriate limit yields the corresponding elegant Laguerre-Gaussian beam. From the spectral representation of the elegant Lauguerre–Gaussian wave we determine the first three orders of nonparaxial corrections for the corresponding paraxial elegant Laguerre–Gaussian beam.

© 2004 Optical Society of America

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References

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1971 (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

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Equations (32)

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Un,mr=Fn,mr,zexp±imϕ,
r2-m2r2+2z2+k2Fn,mr,z=-ScsΘmr2nδrrδz-zcs,
r2=2r2+1rr,
Θm=rm1rrm
Fn,mr,z=0F¯n,mα,zJmαrα dα,
F¯n,mα,z=0Fn,mr,zJmαrr dr,
Fn,mr,z=0-1n-α2n+miScs2β×expiβz-zcsJmαrα dα,
Fn,m0r,z=iScs2kexpikz-zcs×0-1n-α2n+m×exp-iα22kz-zcsJmαrα dα,
0α2n+m exp-p2α2Jmαrα dα=n!2p-2n+m+2r2pmLnmr24p2exp-r24p2,
Fn,m0r,z=-1n+miScs2k×expikz-zcsn!2p-2n+m+2×r2pmLnmr24p2exp-r24p2.
Fn,m0r,0=-1n+m22n+mn!×rw0mLnmr2w02exp-r2w02,
zcs=ikw02/2=izR,
Scs=-i2zRw02n+m exp-kzR,
Fn,m0r,z=-1n+m22n+mn! expikz×h2n+m+2vm/2Lnmvexp-v,
h=hz=1+iz/zR-1/2,
v=vr,z=h2r2/w02.
Un,mr=-1n exp±imϕzR×exp-kzRw02n+m0-α2n+mβ-1×expiβz-izRJmαrα dα.
r2+2z2+k2Gr,z=-Scsδrrδz-izR,
Gr,z=Scs expikR/2R,
Θmr2-r2Θm=-m2r2Θm,
r2-m2r2+2z2+k2Θmr2nGr,z=-ScsΘmr2nδrrδz-izR.
Un,mr=-izRw02n+m×exp-kzRΘmr2n×expikR/Rexp±imϕ.
Un,mr12expikz±imϕw02n+m+2×0-1n-α2n+m×exp-iα22kz-izR×j=03G2jα,zkw02jJmrαα dα,
G0α,z=1,
G2α,z=w02α22-w04α416h2,
G4α,z=3w04α48-w06α616h2+w08α8512h4,
G6α,z=5w06α616-15w08α8256h2+3w010α101024h4-w012α126×4096h6.
Un,mr-1n+m22n+m expikz±imϕ×h2n+m+2vm/2×exp-vj=03hkw02jfn,m2j,
fn,m0=n!Lnmv,
fn,m2=2n+1!Ln+1mv-n+2!Ln+2mv,
fn,m4=6n+2!Ln+2mv-4n+3!Ln+3mv+12n+4!Ln+4mv,
fn,m6=20n+3!Ln+3mv-15n+4!Ln+4mv+3n+5!Ln+5mv-16n+6!Ln+6mv.

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